Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 173430, 15 pages doi:10.1155/2011/173430 Research Article An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces Ming Tian College of Science, Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Ming Tian, tianming1963@126.com Received December 2010; Accepted 13 February 2011 Academic Editor: Shusen Ding Copyright q 2011 Ming Tian This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces We proved strong convergence theorems of the sequence generated by our proposed schemes Introduction Let H be a real Hilbert space and C a closed convex subset of H, and let φ be a bifunction of C × C into R, where R is the set of real numbers The equilibrium problem for φ : C × C → R is to find x ∈ C such that EP : φ x, y ≥ ∀y ∈ C 1.1 denoted the set of solution by EP φ Given a mapping T : C → H, let φ x, y Tx, y − x for all x, y ∈ C, then z ∈ EP φ if and only if Tz, y − z ≥ for all y ∈ C, that is, z is a solution of the variational inequality Numerous problems in physics, optimizations, and economics reduce to find a solution of 1.1 Some methods have been proposed to solve the equilibrium problem, see, for instance, 1, A mapping T of C into itself is nonexpansive if Tx−Ty ≤ x−y , for all x, y ∈ C The set of fixed points of T is denoted by F T In 2007, Plubtieng and Punpaeng , S Takahashi and W Takahashi , and Tada and W Takahashi considered iterative methods for finding an element of EP φ ∩ F T Journal of Inequalities and Applications Recall that an operator A is strongly positive if there exists a constant γ > with the property Ax, x ≥ γ x , ∀x ∈ H 1.2 In 2006, Marino and Xu introduced the general iterative method and proved that for a given x0 ∈ H, the sequence {xn } is generated by the algorithm xn αn γ f xn I − αn A Txn , n ≥ 0, 1.3 where T is a self-nonexpansive mapping on H, f is a contraction of H into itself with β ∈ 0, and {αn } ⊂ 0, satisfies certain conditions, and A is a strongly positive bounded linear operator on H and converges strongly to a fixed-point x∗ of T which is the unique solution to the following variational inequality: γ f − A x∗ , x − x∗ ≤ 0, for x ∈ F T , and is also the optimality condition for some minimization problem A mapping S : C → H is said to be k-strictly pseudocontractive if there exists a constant k ∈ 0, such that Sx − Sy ≤ x−y k I−S x− I−S y , ∀x, y ∈ C 1.4 Note that the class of k-strict pseudo-contraction strictly includes the class of nonexpansive mapping, that is, S is nonexpansive if and only if S is 0-srictly pseudocontractive; it is also said to be pseudocontractive if k Clearly, the class of k-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions The set of fixed points of S is denoted by F S Very recently, by using the general approximation method, Qin et al obtained a strong convergence theorem for finding an element of F S On the other hand, Ceng et al proposed an iterative scheme for finding an element of EP φ ∩ F S and then obtained some weak and strong convergence theorems Based on the above work, Y Liu introduced two iteration schemes by the general iterative method for finding an element of EP φ ∩ F S In 2001, Yamada 10 introduced the following hybrid iterative method for solving the variational inequality: xn Txn − μλn F Txn , n ≥ 0, 1.5 where F is k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0, < μ < 2η/k2 , then he proved that if {λn } satisfyies appropriate conditions, the {xn } generated by 1.5 converges strongly to the unique solution of variational inequality F x, x − x ≥ 0, ∀x ∈ Fix T , x ∈ Fix T 1.6 Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of EP φ ∩ F S , where S : C → H is a k-strictly pseudocontractive non-self mapping, and then obtained two strong convergence theorems Journal of Inequalities and Applications Preliminaries Throughout this paper, we always assume that C is a nonempty closed convex subset of a x to indicate that the sequence {xn } converges weakly to Hilbert space H We write xn x xn → x implies that {xn } converges strongly to x For any x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that x − PC x ≤ x − y , ∀y ∈ C 2.1 Such a PC x is called the metric projection of H onto C It is known that PC is nonexpansive Furthermore, for x ∈ H and u ∈ C, u pc x, ⇔ x − u, u − y ≥ 0, for all y ∈ C It is widely known that H satisfies Opial’s condition 11 , that is, for any sequence {xn } with xn x, the inequality lim inf xn − x < lim inf xn − y , n→∞ n→∞ 2.2 holds for every y ∈ H with y / x In order to solve the equilibrium problem for a bifunction φ : C × C → R, let us assume that φ satisfies the following conditions: A1 φ x, x 0, for all x ∈ C, A2 φ is monotone, that is, φ x, y φ y, x ≤ 0, for all x, y ∈ C, A3 For all x, y, z ∈ C limφ tz t↓0 − t x, y ≤ φ x, y ; 2.3 A4 For each fixed x ∈ C, the function y → φ x, y is convex and lower semicontinuous Let us recall the following lemmas which will be useful for our paper Lemma 2.1 see 12 Let φ be a bifunction from C × C into R satisfying (A1), (A2),(A3) and (A4) then, for any r > and x ∈ H, there exists z ∈ C such that y − z, z − x ≥ 0, r φ z, y Further, if Tr x {z ∈ C : φ z, y ∀y ∈ C 2.4 1/r y − z, z − x ≥ 0, ∀y ∈ C}, then the following hold: Tr is single-valued, Tr is firmly nonexpansive, that is, Tr x − Tr y F Tr ≤ Tr x − Tr y, x − y , EP φ , EP φ is nonempty, closed and convex ∀x, y ∈ H; 2.5 Journal of Inequalities and Applications Lemma 2.2 see 13 If S : C → H is a k-strict pseudo-contraction, then the fixed-point set F S is closed convex, so that the projection PF S is well difened Lemma 2.3 see 14 Let S : C → H be a k-strict pseudo-contraction Define T : C → H by Tx λx − λ Sx for each x ∈ C, then, as λ ∈ k, , T is nonexpansive mapping such that F T F S Lemma 2.4 see 15 In a Hilbert space H, there holds the inequality x y ≤ x 2 y, x y , ∀x, y ∈ H 2.6 Lemma 2.5 see 16 Assume that {an } is a sequence of nonnegative real numbers such that an ≤ − γn an γ n δn , n ≥ 0, 2.7 where {γn } is a sequence in (0,1) and {δn } is a sequence in Ê, such that i ∞ n γn ∞, ii lim supn → ∞ δn ≤ or Then limn → ∞ an ∞ n |δn γn | < ∞ Main Results Throughout the rest of this paper, we always assume that F is a L-lipschitzian continuous and η-strongly monotone operator with L, η > and assume that < μ < 2η/L2 τ μ η − μL2 /2 Let {Tλn } be mappings defined as Lemma 2.1 Define a mapping Sn : C → H by Sn x βn x − βn Sx, for all x ∈ C, where βn ∈ k, , then, by Lemma 2.3, Sn is nonexpansive We consider the mapping Gn on H defined by Gn x I − αn μF Sn Tλn x, x ∈ H, n ∈ N, 3.1 where αn ∈ 0, By Lemmas 2.1 and 2.3, we have Gn x − Gn y ≤ − αn τ ≤ − αn τ Tλn x − Tλn y x−y 3.2 It is easy to see that Gn is a contraction Therefore, by the Banach contraction principle, F Gn has a unique fixed-point xn ∈ H such that F xn F I − αn μF Sn Tλn xn 3.3 Journal of Inequalities and Applications F For simplicity, we will write xn for xn provided no confusion occurs Next, we prove that the sequence {xn } converges strongly to a q ∈ F S ∩ EP φ which solves the variational inequality Fq, p − q ≥ 0, Equivalently, q PF ∀p ∈ F S ∩ EP φ 3.4 I − μF q S ∩EP φ Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction from C × C into R satisfying (A1), (A2), (A3), and (A4) Let S : C → H be a kstrictly pseudocontractive nonself mapping such that F S ∩ EP φ / φ Let F : H → H be an L-Lipschitzian continuous and η-strongly monotone operator on H with L, η > and < μ < 2η/L2 , τ μ η − μL2 /2 Let {xn } be asequence generated by φ un , y y − un , un − xn ≥ 0, ∀y ∈ C, λn − βn Sun , yn βn un xn I − αn μF yn , ∀n ∈ N, 3.5 where un Tλn xn , yn Sn un , and {λn } ⊂ 0, ∞ satisfy lim infn → ∞ λn > if {αn } and {βn } satisfy the following conditions: i {αn } ⊂ 0, , limn → ∞ αn 0, ii ≤ k ≤ βn ≤ λ < and limn → ∞ βn λ, then {xn } converges strongly to a point q ∈ F S ∩ EP φ which solves the variational inequality 3.4 Proof First, take p ∈ F S ∩ EP φ Since un n ∈ N, we have un − p Then, since Sn p Tλn xn and p Tλn p, from Lemma 2.1, for any Tλn xn − Tλn p ≤ xn − p 3.6 p, we obtain that yn − p Sn un − Sn p ≤ un − p ≤ xn − p 3.7 Further, we have xn − p −αn μFp ≤ αn −μF p It follows that xn − p ≤ μF p /τ I − μαn F yn − I − μαn F p − αn τ yn − p 3.8 Journal of Inequalities and Applications Hence, {xn } is bounded, and we also obtain that {un } and {yn } are bounded Notice that un − yn ≤ un − xn xn − yn 3.9 αn −μFyn un − xn By Lemma 2.1, we have un − p Tλn xn − Tλn p 2 xn − p ≤ xn − p, un − p un − p 3.10 − un − xn It follows that un − p ≤ xn − p − xn − un 3.11 Thus, from Lemma 2.4, 3.7 , and 3.11 , we obtain that xn − p αn −μFp I − μαn F yn − I − μαn F p ≤ − αn τ yn − p 2αn −μFp, xn − p ≤ − αn τ un − p 2αn −μFp, xn − p ≤ − αn τ 2 xn − p − 2αn τ αn τ − − αn τ ≤ xn − p 2 2 2 2αn −μFp xn − p xn − p xn − un αn τ − xn − un 2αn − μFp xn − p xn − p − − αn τ xn − un 2αn −μFp xn − p 3.12 It follows that − αn τ xn − un ≤ αn τ xn − p 2αn μFp xn − p 3.13 Since αn → 0, therefore lim xn − un 3.14 lim un − yn 3.15 n→∞ From 3.9 , we derive that n→∞ Journal of Inequalities and Applications Define T : C → H by Tx by Lemma 2.3 We note that λx − λ Sx, then T is nonexpansive with F T yn − un ≤ λ − βn un − Sun Tun − un ≤ Tun − yn yn − un F S 3.16 So by 3.15 and βn → λ, we obtain that lim Tun − un n→∞ 3.17 Since {un } is bounded, so there exists a subsequence {uni } which converges weakly to q Next, we show that q ∈ F S ∩ EP φ Since C is closed and convex, C is weakly closed So q and q / Tq, it we have q ∈ C Let us show that q ∈ F S Assume that q ∈ F T , Since uni follows from the Opial’s condition that lim inf uni − q < lim inf uni − Tq n→∞ n→∞ ≤ lim inf uni − Tuni n→∞ Tuni − Tq 3.18 ≤ lim inf uni − q n→∞ This is a contradiction So, we get q ∈ F T and q ∈ F S Next, we show that q ∈ EP φ Since un Tλn xn , for any y ∈ C, we obtain φ un , y y − un , un − xn ≥ λn 3.19 From A2 , we have y − un , un − xn ≥ φ y, un λn 3.20 Replacing n by ni , we have y − uni , uni − xni λni ≥ φ y, uni 3.21 Since uni − xni /λni → and uni q, it follows from A4 that ≥ φ y, q , for all ty − t q for all t ∈ 0, and y ∈ C, then we have zt ∈ C and hence y ∈ C Let zt φ zt , q ≤ Thus, from A1 and A4 , we have φ zt , zt ≤ tφ zt , y − t φ zt , q ≤ tφ zt , y , 3.22 Journal of Inequalities and Applications and hence ≤ φ zt , y From A3 , we have ≤ φ q, y for all y ∈ C and hence q ∈ EP φ Therefore, q ∈ F S ∩ EP φ On the other hand, we note that xn − q −αn μFq I − μαn F yn − I − μαn F q 3.23 Hence, we obtain xn − q −αn μFq, xn − q I − μαn F yn − I − μαn F q, xn − q ≤ αn −μFq, xn − q − αn τ xn − q 3.24 It follows that xn − q ≤ −μFq, xn − q τ 3.25 This implies that xn − q ≤ −μFq, xn − q τ 3.26 xni − q ≤ −μFq, xni − q τ 3.27 In particular, Since xni q, it follows from 3.27 that xni → q as i → ∞ Next, we show that q solves the variational inequality 3.4 As a matter of fact, we have xn I − αn μF yn 3.28 I − αn μF Sn Tλn xn , and we have μFxn − αn I − Sn Tλn xn − μαn Fxn − FSn Tλn xn 3.29 Hence, for p ∈ F S ∩ EP φ , μF xn , xn − p − αn − αn I − Sn Tλn xn − μαn Fxn − FSn Tλn xn , xn − p I − Sn Tλn xn − I − Sn Tλn p, xn − p μ Fxn − FSn Tλn xn , xn − p 3.30 Journal of Inequalities and Applications Since I − Sn Tλn is monotone i.e., x − y, I − Sn Tλn x − I − Sn Tλn y ≥ 0, for all x, y ∈ H This is due to the nonexpansivity of Sn Tλn Now replacing n in 3.30 with ni and letting i → ∞, we obtain μF q, q − p lim μFxni , xni − p i→∞ ≤ lim μ Fxni − FSn Tλn xni , xni − p i→∞ 3.31 That is, q ∈ F S ∩EP φ is a solution of 3.4 To show that the sequence {xn } converges strongly to q, we assume that xnk → x Similiary to the proof above, we derive x ∈ F S ∩ EP φ Moreover, it follows from the inequality 3.31 that μF q, q − x ≤ 3.32 μF x, x − q ≤ 3.33 Interchange q and x to obtain Adding up 3.32 and 3.33 yields μη Hence, q q−x ≤ q − x, μF q − μF x ≤ 3.34 x, and therefore xn → q as n → ∞, I − μF q − q, q − p ≥ 0, ∀p ∈ F S ∩ EP φ 3.35 This is equivalent to the fixed-point equation PF S ∩EP φ I − μF q q 3.36 Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction from C × C into R satisfying (A1), (A2), (A3) and (A4) Let S : C → H be a k-strictly pseudocontractive nonself mapping such that F S ∩EP φ / φ Let F : H → H be an L-Lipschitzian continuous and η-strongly monotone operator on H with L, η > Suppose that < μ < 2η/L2 , τ μ η − μL2 /2 Let {xn } and {un } be sequences generated by x1 ∈ H and φ un , y xn y − un , un − xn ≥ 0, λn − βn Sun , yn βn un I − αn μF yn , ∀y ∈ C, ∀n ∈ N, 3.37 10 Journal of Inequalities and Applications where un Tλn xn , yn Sn un if {αn },{βn }, and {λn } satisfy the following conditions: i {αn } ⊂ 0, , limn → ∞ αn ∞ n 0, αn ii ≤ k ≤ βn ≤ λ < and limn → ∞ βn ∞ n λ, ∞ n iii {λn } ∈ 0, ∞ , limn → ∞ λn > and ∞ n ∞, |βn |λn 1 |αn − αn | < ∞, − βn | < ∞, − λn | < ∞, then {xn } and {un } converge strongly to a point q ∈ F S ∩EP φ which solves the variational inequality 3.4 Proof We first show that {xn } is bounded Indeed, pick any p ∈ F S ∩ EP φ to derive that xn −p −αn μFp I − μαn F yn − I − μαn F p ≤ αn −μF p ≤ − αn τ − αn τ xn − p xn − p αn −μF p 3.38 By induction, we have xn − p ≤ max x1 − p , − μF p τ , ∀n ∈ N, 3.39 and hence {xn } is bounded From 3.6 and 3.7 , we also derive that {un } and {yn } are bounded Next, we show that xn − xn → We have xn − xn I − αn μF yn − I − αn−1 μF yn−1 I − αn μF yn − I − αn μF yn−1 I − αn μF yn−1 − I − αn−1 μF yn−1 ≤ − αn τ yn − yn−1 |αn − αn−1 | μFyn−1 ≤ − αn τ yn − yn−1 K|αn − αn−1 |, 3.40 where K sup μFyn : n ∈ N < ∞ 3.41 On the other hand, we have yn − yn−1 Sn un − Sn−1 un−1 ≤ Sn un − Sn un−1 ≤ un − un−1 Sn un−1 − Sn−1 un−1 Sn un−1 − Sn−1 un−1 3.42 Journal of Inequalities and Applications From un Tλn xn 1 and un λn y − un , un 1 − xn ≥ 0, 1 y − un , un − xn ≥ 0, λn φ un , y Putting y Tλn xn , we note that φ un , y 11 un in 3.43 and y φ un , un φ un , un un ∀y ∈ C, 3.43 ∀y ∈ C 3.44 in 3.44 , we have un − un , un − xn ≥ 0, λn 1 un − un , un − xn ≥ λn 3.45 So, from A2 , we have un un − xn un − xn − λn λn − un , ≥0, 3.46 and hence un − un , un − un un − xn − λn un λn 1 − xn ≥ 3.47 Since limn → ∞ λn > 0, without loss of generality, let us assume that there exists a real number a such that λn > a > for all n ∈ N Thus, we have un − un ≤ un − un , xn 1 − xn ≤ un un where M0 1 − un xn − un ≤ xn − xn |λn a 1− − xn λn λn 1− λn λn un un − xn 1 − xn 3.48 − λn |M0 , sup{ un − xn : n ∈ N} Next, we estimate Sn un−1 − Sn−1 un−1 Notice that Sn un−1 − Sn−1 un−1 βn un−1 − βn Sun−1 − βn−1 un−1 − βn−1 Sun−1 ≤ βn − βn−1 un−1 − Sun−1 3.49 From 3.48 , 3.49 , and 3.42 , we obtain that yn − yn−1 ≤ xn − xn−1 M0 |λn − λn−1 | a βn − βn−1 un−1 − Sun−1 ≤ xn − xn−1 |λn − λn−1 |M1 βn − βn−1 M1 , 3.50 12 Journal of Inequalities and Applications where M1 is an appropriate constant such that M0 a M1 ≥ un−1 − Sun−1 , ∀n ∈ N 3.51 From 3.41 and 3.50 , we obtain xn − xn ≤ K|αn − αn−1 | − αn τ ≤ − αn τ xn − xn−1 where M xn − xn−1 |λn − λn−1 |M1 M |αn − αn−1 | |λn − λn−1 | βn − βn−1 M1 βn − βn−1 , 3.52 max K, M1 Hence, few by Lemma 2.5, we have lim xn n→∞ − xn 3.53 From 3.48 and 3.50 , |λn − λn−1 | → and |βn − βn−1 | → 0, we have lim un n→∞ − un lim yn 0, n→∞ − yn 3.54 Since xn I − αn μF yn , 3.55 it follows that xn − yn ≤ xn − xn xn − xn xn − yn 3.56 αn −μFyn From αn → and 3.53 , we have lim xn − yn n→∞ 3.57 For p ∈ F S ∩ EP φ , we have un − p Tλn xn − Tλn p xn − p 2 ≤ xn − p, un − p un − p 3.58 − un − xn This implies that un − p ≤ xn − p − un − xn 3.59 Journal of Inequalities and Applications 13 Then, from 3.7 and 3.59 , we derive that xn −p −μαn Fp I − μαn F yn − I − μαn F p ≤ − αn τ ≤ un − p ≤ xn − p Since αn → 0, xn − xn yn − p α2 −μFp n α2 −μFp n 2αn −μFp α2 −μFp n yn − p yn − p 2αn −μFp − xn − un 2 2αn −μFp 3.60 yn − p → 0, we have lim xn − un n→∞ 3.61 From 3.57 and 3.61 , we obtain that un − yn ≤ un − xn Define T : C → H by Tx by Lemma 2.3 Notice that λx xn − yn → 0, as n → ∞ − λ Sx, then T is nonexpansive with F T yn − un Tun − un ≤ Tun − yn ≤ λ − βn un − Sun yn − un 3.62 F S 3.63 By 3.62 and βn → λ, we obtain that lim Tun − un n→∞ 3.64 Next, we show that lim supn → ∞ μFq, q − xn ≤ 0, where q PF S ∩EP φ I − μF q is a unique solution of the variational inequality 3.4 Indeed, take a subsequence {xni } of {xn } such that lim μFq, q − xni i→∞ lim sup μFq, q − xn n→∞ 3.65 Since {xni } is bounded, there exists a subsequence {xnij } of {uni } which converges weakly to w Without loss of generality, we can assume that uni w From 3.61 and 3.64 , we w and Tuni w By the same argument as in the proof of Theorem 3.1, we have obtain xni w ∈ F S ∩ EP φ Since q PF S ∩EP φ I − μF q, it follows that lim sup μFq, q − xn n→∞ μFq, q − w ≤ 3.66 14 Journal of Inequalities and Applications From xn xn −q −αn μFq −q ≤ I − μαn F yn − I − μαn F q, we have I − μαn F yn − I − μαn F q ≤ − αn τ xn − q 2 2αn −μFq, xn 2αn −μFq, xn 1 −q −q 3.67 This implies that xn −q ≤ − 2αn τ αn τ − 2αn τ xn − q αn τ − 2αn τ xn − q 2αn τ − γn xn − q 2 xn − q 2αn −μFq, xn xn − q αn τ ∗ M 2τ −q 2αn −μFq, xn −μFq, xn τ −q 3.68 −q γ n δn , where M∗ sup{ xn −q : n ∈ N}, γn 2αn τ, and δn αn τ /2τ M∗ 1/τ −μFq, xn −q ∞ It is easy to see that γn → 0, n γn ∞, and lim supn → ∞ δn ≤ by 3.66 Hence by Lemma 2.5, the sequence {xn } converges strongly to q Acknowledgments M 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